M35b The Heun Hyperradical: Degree-Six Polynomials, Grand Unification, and Lambert's Orbital Equation

M35b constructs the Heun hyperradical H6, a solving operation for generic degree‑6 polynomials, located at rank R = 5/2 (Genus‑2 / Heun layer). 1. Position in the hyperradical ladderM35b is the middle element of the ladder:5 3/2 AGM H5 (Hermite 1858, M29a - general insight)6 5/2 Heun H6 (M35b)7 7/2 ISHE H7 (M35)Key statement:Solvability is not lost after degree 4. It shifts to higher operational ranks. 2. Core mathematical claimGeneric degree‑6 polynomials are solvable at rank R = 5/2 via the Heun hyperradical H6. Mechanism:Galois complexity → genus 5 → intermediate structure → R = 2.5 3. Construction mechanismThe solving chain is:General sextic→ Bring–Harriot reduction→ intermediate equations of degree ≤ 5→ solved using AGM hyperradical H5→ lifted via Heun structure to full solution So structurally:H6 = H5 + Heun layerImportant property:H6 is not independent: it bootstraps on H5. 4. Nature of the Heun layerThe Heun level (R = 5/2) is characterised by:- genus‑2 / genus‑5 (depending on perspective)- Heun differential equations (generalization of hypergeometric)- multi‑branch monodromy structureInterpretation:AGM → single elliptic curveHeun → interacting elliptic structures / higher genusThus:Heun = next complexity level of algebraic solvability after elliptic (AGM). 5. Bootstrap structure (central insight)M35b establishes:H5 → solves degree 5H6 → requires H5H7 → requires H6So:Hyperradicals form a recursive hierarchy.This is the first place in the corpus where: solvers are not independent, but form a nested computational chain. 6. Conceptual shiftClassical statement:Degree ≥ 5 → unsolvable by radicalsOperational reinterpretation:Each degree requires a fractional and higher rank.Specifically:degree 5 → first non-radical → first hyperradical (AGM)degree 6 → next obstruction → HeunSo:unsolvability = crossing a rank boundary 7. Structural role in the programmeM25 → builds analytic structure (genus / Picard–Fuchs)M29 → maps Galois group → rankM35b → constructs the solver at that rankM35 → completes next level (ISHE)Thus:M35b = arithmetic realization of the Heun layer.8. Orbital mechanics, GUT and predictions The same object appears in two apparently distant settings: GUT:degree-6 symmetry-breaking equation→ solved by H6 Orbital mechanics:degree-6 critical-orbit equation→ solved by H6 The common reason is the rank: both problems cross the same Heun seam. They are not identical physical systems, but they occupy the same operational layer. In the GUT reading, the CatHeun Hermit unittau_CH = exp(W_0(i*pi)) is the operational marker of the Heun layer. The document identifies this unit with a GUT scaleE_GUT ≈ 1.86 × 10^16 GeV and records the associated proton lifetime estimatetau_p ≈ 5.4 × 10^36 yr.